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Competition models with diffusion, analytic semigroups, and inertial manifolds
Author(s) -
Nguyen Thieu Huy,
Bui XuanQuang
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5281
Subject(s) - mathematics , lipschitz continuity , semigroup , inertial frame of reference , operator (biology) , competition (biology) , mathematical analysis , nonlinear system , analytic semigroup , population , space (punctuation) , banach space , pure mathematics , ecology , physics , biology , biochemistry , chemistry , demography , linguistics , philosophy , repressor , quantum mechanics , sociology , transcription factor , gene
Motivated by a two‐specie competition model with cross‐diffusion in population ecology with time‐dependent environmental capacity, we study the semilinear parabolic evolution equation of the form x ˙ + A x = f ( t , x ) where the linear operator − A generates an analytic semigroup. Our main task is to prove the existence of inertial manifolds for mild solutions to such an evolution equation under the conditions that the linear partial differential operator − A has spectral gap being sufficiently large, and the nonlinear term f satisfies ϕ ‐Lipschitz condition, ie, ‖ f ( t , x ) − f ( t , y )‖ ≤ ϕ ( t )‖ A θ ( x − y )‖ for ϕ belonging to some admissible space. We then apply the obtained result to study asymptotic behavior of the above‐mentioned competition model.